On Orbits of SL(2,image)+ and Values of Binary Quadratic Forms on Positive Integral Pairs Original Research Article

We consider actions of SL(2,image) and SL(2,image)+ (semigroup of matrices with nonnegative integral entries) on the projective space image and on image×image. Results are obtained on orbit-closures under these actions and they are applied to describe a class of binary quadratic forms Q such that the sets Q(image2) or Q(image+2) are dense in image. We prove also a result generalising a theorem of Troessaert and Valette (On values at integer points of some irrational quadratic forms, preprint) associating to certain binary quadratic forms Q rational quadratic forms Q′ such that the closure of Q(image2) contains Q′(image2).